Definition:Canonical Mapping from Coproduct to Product

Definition

Let $C$ be a category with zero morphisms.

Let $a, b$ be objects of $C$.

Assume they have a coproduct $(a \sqcup b, i_1, i_2)$ and a product $(a \times b, p_1, p_2)$.

Definition 1

Let:

$j_1 : a \to a \times b$ be the unique morphism such that:

• $p_1 \circ j_1 = 1 : a \to a$
• $p_2 \circ j_1 = 0 : a \to b$

$j_2 : b \to a \times b$ be the unique morphism such that:

• $p_1 \circ j_2 = 0 : b \to a$
• $p_2 \circ j_2 = 1 : b \to b$

The canonical mapping from $a \sqcup b$ to $a \times b$ is the unique morphism $r : a \sqcup b \to a \times b$ such that:

$r \circ i_1 = j_1$

$r \circ i_2 = j_2$

Definition 2

Let:

$q_1 : a \sqcup b \to a$ be the unique morphism such that:

• $q_1 \circ j_1 = 1 : a \to a$
• $q_1 \circ j_2 = 0 : b \to a$

$q_2 : a \sqcup b \to b$ be the unique morphism such that:

• $q_2 \circ j_1 = 0 : a \to a$
• $q_2 \circ j_2 = 1 : b \to a$

The canonical mapping from $a \sqcup b$ to $a \times b$ is the unique morphism $r : a \sqcup b \to a \times b$ such that:

$p_1 \circ r = q_1$

$p_2 \circ r = q_2$

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Also see

• Equivalence of Definitions of Canonical Mapping from Coproduct to Product
• Definition:Natural Transformation from Coproduct Functo to Product Functor